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research
Besov regularity of solutions to the p-Poisson equation
Authors
Stephan Dahlke
Lars Diening
+3Â more
Christoph Hartmann
Benjamin Scharf
Markus Weimar
Publication date
19 August 2014
Publisher
View
on
arXiv
Abstract
In this paper, we study the regularity of solutions to the
p
p
p
-Poisson equation for all
1
<
p
<
∞
1<p<\infty
1
<
p
<
∞
. In particular, we are interested in smoothness estimates in the adaptivity scale
B
Ï„
σ
(
L
Ï„
(
Ω
)
)
B^\sigma_{\tau}(L_{\tau}(\Omega))
B
Ï„
σ
​
(
L
Ï„
​
(
Ω
))
,
1
/
Ï„
=
σ
/
d
+
1
/
p
1/\tau = \sigma/d+1/p
1/
Ï„
=
σ
/
d
+
1/
p
, of Besov spaces. The regularity in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear approximation methods. It turns out that, especially for solutions to
p
p
p
-Poisson equations with homogeneous Dirichlet boundary conditions on bounded polygonal domains, the Besov regularity is significantly higher than the Sobolev regularity which justifies the use of adaptive algorithms. This type of results is obtained by combining local H\"older with global Sobolev estimates. In particular, we prove that intersections of locally weighted H\"older spaces and Sobolev spaces can be continuously embedded into the specific scale of Besov spaces we are interested in. The proof of this embedding result is based on wavelet characterizations of Besov spaces.Comment: 45 pages, 2 figure
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Last time updated on 29/10/2017